Last Time

1. Last Time • Drawing lines • Inside/Outside tests for polygons (c) 2002 University of Wisconsin, CS559

2. Today • Drawing Polygons (c) 2002 University of Wisconsin, CS559

3. What is inside - 1? • Easy for simple polygons - no self intersections or holes • OpenGL requires these. Undefined for other cases • OpenGL also requires convex polygons • For general polygons, three rules are possible: • Non-exterior rule: A point is inside if every ray to infinity intersects the polygon • Non-zero winding number rule: Draw a ray to infinity that does not hit a vertex, if the number of edges crossing in one direction is not equal to the number crossing the other way, the point is inside • Parity rule: Draw a ray to infinity and count the number or edges that cross it. If even, the point is outside, if odd, it’s inside (c) 2002 University of Wisconsin, CS559

4. Inside/Outside Rules Polygon Non-exterior Non-zero Winding No. Parity (c) 2002 University of Wisconsin, CS559

5. What is inside - 2? • Assume sampling with an array of spikes • If spike is inside, pixel is inside (c) 2002 University of Wisconsin, CS559

6. What is inside - 2? • Assume sampling with an array of spikes • If spike is inside, pixel is inside (c) 2002 University of Wisconsin, CS559

7. Ambiguous Case • Ambiguous cases: What if a pixel lies on an edge? • Problem because if two polygons share a common edge, we don’t want pixels on the edge to belong to both • Ambiguity would lead to different results if the drawing order were different • Rule: if (x+, y+) is in, (x,y) is in • What if a pixel is on a vertex? Does our rule still work? (c) 2002 University of Wisconsin, CS559

8. Ambiguous Case 1 • Rule: • On edge? If (x+, y+) is in, pixel is in • Which pixels are colored? (c) 2002 University of Wisconsin, CS559

9. Ambiguous Case 1 • Rule: • Keep left and bottom edges • Assuming y increases in the up direction • If rectangles meet at an edge, how often is the edge pixel drawn? (c) 2002 University of Wisconsin, CS559

10. Ambiguous Case 2 (c) 2002 University of Wisconsin, CS559

11. Ambiguous Case 2 ? ? or ? ? (c) 2002 University of Wisconsin, CS559

12. Really Ambiguous • We will accept ambiguity in such cases • The center pixel may end up colored by one of two polygons in this case • Which two? 1 6 2 5 3 4 (c) 2002 University of Wisconsin, CS559

13. Exploiting Coherence • When filling a polygon • Several contiguous pixels along a row tend to be in the polygon - a span of pixels • Scanline coherence • Consider whole spans, not individual pixels • The pixels required don’t vary much from one span to the next • Edge coherence • Incrementally update the span endpoints (c) 2002 University of Wisconsin, CS559

14. Sweep Fill Algorithms • Algorithmic issues: • Reduce to filling many spans • Which edges define the span of pixels to fill? • How do you update these edges when moving from span to span? • What happens when you cross a vertex? (c) 2002 University of Wisconsin, CS559

15. Spans • Process - fill the bottom horizontal span of pixels; move up and keep filling • Have xmin, xmaxfor each span • Define: • floor(x): largest integer < x • ceiling(x): smallest integer >=x • Fill from ceiling(xmin) up to floor(xmax) • Consistent with convention (c) 2002 University of Wisconsin, CS559

16. Algorithm • For each row in the polygon: • Throw away irrelevant edges • Obtain newly relevant edges • Fill span • Update current edges • Issues: • How do we update existing edges? • When is an edge relevant/irrelevant? • All can be resolved by referring to our convention about what polygon pixel belongs to (c) 2002 University of Wisconsin, CS559

17. Updating Edges • Each edge is a line of the form: • Next row is: • So, each current edge can have it’s x position updated by adding a constant stored with the edge • Other values may also be updated, such as depth or color information (c) 2002 University of Wisconsin, CS559

18. When are Edges Relevant (1) • Use figures and convention to determine when edge is irrelevant • For y<ymin and y>=ymax of edge • Similarly, edge is relevant when y>=ymin and y<ymax of edge • What about horizontal edges? • m’ is infinite (c) 2002 University of Wisconsin, CS559

19. When are Edges Relevant (2) Convex polygon: Always only two edges active 2 1,2 1 1,3 3 4 3,4 (c) 2002 University of Wisconsin, CS559

20. When are Edges Relevant (3) Horizontal edges? 2 2? 1 3 1,3 4? 4 (c) 2002 University of Wisconsin, CS559

21. Maintain a list of active edges in case there are multiple spans of pixels - known as Active Edge List. For each edge on the list, must know: x-value, maximum y value of edge, m’ Maybe also depth, color… Keep all edges in a table, indexed by minimum y value - Edge Table For row = min to row=max AEL=append(AEL, ET(row)); remove edges whose ymax=row sort AEL by x-value fill spans update each edge in AEL Sweep Fill Details (c) 2002 University of Wisconsin, CS559

22. Edge Table Row: 6 6 5 5 4 4 0 6 4 3 3 2 2 1 2 0 4 6 0 6 1 6 0 6 1 2 3 4 5 6 ymax xmin 1/m (c) 2002 University of Wisconsin, CS559

23. Active Edge List (shown just before filling each row) Row: 6 6 5 4 0 6 6 0 6 5 4 4 0 6 6 0 6 4 3 2 0 4 6 0 6 3 2 2 0 4 6 0 6 2 1 2 0 4 6 0 6 1 6 0 6 1 2 3 4 5 6 ymax x 1/m (c) 2002 University of Wisconsin, CS559

24. Edge Table Row: 6 6 5 5 4 4 3 3 2 2 1 2 1 5 6 -1 5 1 6 0 6 1 2 3 4 5 6 ymax xmin 1/m (c) 2002 University of Wisconsin, CS559

25. Active Edge List Row: 6 6 5 5 4 3 -1 5 5 1 5 4 3 4 1 5 4 -1 5 3 2 3 1 5 5 -1 5 2 1 2 1 5 6 -1 5 1 6 0 6 1 2 3 4 5 6 ymax x 1/m (c) 2002 University of Wisconsin, CS559

26. Comments • Sort is quite fast, because AEL is usually almost in order • OpenGL limits to convex polygons, meaning two and only two elements in AEL at any time, and no sorting • Can generate memory addresses (for pixel writes) efficiently • Does not require floating point - next slide (c) 2002 University of Wisconsin, CS559

27. Avoiding Floating Point • For edge, m=x/y, which is a rational number • View x as xi+xn/y, with xn<y. Store xi and xn • Then x->x+m’ is given by: • xn=xn+x • if (xn>=y) { xi=xi+1; xn=xn- y } • Advantages: • no floating point • can tell if x is an integer or not, and get floor(x) and ceiling(x) easily, for the span endpoints (c) 2002 University of Wisconsin, CS559